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A reconstruction theorem for locally moving groups acting on completely metrizable spaces

Edmund Ben-Ami (2010)

Fundamenta Mathematicae

Let G be a group which acts by homeomorphisms on a metric space X. We say the action of G is locally moving on X if for every open U ⊆ X there is a g ∈ G such that g↾X ≠ Id while g↾(X∖U) = Id. We prove the following theorem: Theorem A. Let X,Y be completely metrizable spaces and let G be a group which acts on X and Y with locally moving actions. If the orbits of the action of G on X are of the second category in X and the orbits of the action of G on Y are of the second category...

An application of Lie groupoids to a rigidity problem of 2-step nilmanifolds

Hamid-Reza Fanaï, Atefeh Hasan-Zadeh (2019)

Mathematica Bohemica

We study a problem of isometric compact 2-step nilmanifolds M / Γ using some information on their geodesic flows, where M is a simply connected 2-step nilpotent Lie group with a left invariant metric and Γ is a cocompact discrete subgroup of isometries of M . Among various works concerning this problem, we consider the algebraic aspect of it. In fact, isometry groups of simply connected Riemannian manifolds can be characterized in a purely algebraic way, namely by normalizers. So, suitable factorization...

Coarse structures and group actions

N. Brodskiy, J. Dydak, A. Mitra (2008)

Colloquium Mathematicae

The main results of the paper are: Proposition 0.1. A group G acting coarsely on a coarse space (X,𝓒) induces a coarse equivalence g ↦ g·x₀ from G to X for any x₀ ∈ X. Theorem 0.2. Two coarse structures 𝓒₁ and 𝓒₂ on the same set X are equivalent if the following conditions are satisfied: (1) Bounded sets in 𝓒₁ are identical with bounded sets in 𝓒₂. (2) There is a coarse action ϕ₁ of a group G₁ on (X,𝓒₁) and a coarse action ϕ₂ of a...

Metric groups, unitary representations and continuous logic

Aleksander Ivanov (2021)

Communications in Mathematics

We describe how properties of metric groups and of unitary representations of metric groups can be presented in continuous logic. In particular we find L ω 1 ω -axiomatization of amenability. We also show that in the case of locally compact groups some uniform version of the negation of Kazhdan’s property (T) can be viewed as a union of first-order axiomatizable classes. We will see when these properties are preserved under taking elementary substructures.

On derivations and crossed homomorphisms

Viktor Losert (2010)

Banach Center Publications

We discuss some results about derivations and crossed homomorphisms arising in the context of locally compact groups and their group algebras, in particular, L¹(G), the von Neumann algebra VN(G) and actions of G on related algebras. We answer a question of Dales, Ghahramani, Grønbæk, showing that L¹(G) is always permanently weakly amenable. Then we show that for some classes of groups (e.g. IN-groups) the homology of L¹(G) with coefficients in VN(G) is trivial. But this is no longer true, in general,...

On extendability of invariant distributions

Bogdan Ziemian (2000)

Annales Polonici Mathematici

In this paper sufficient conditions are given in order that every distribution invariant under a Lie group extend from the set of orbits of maximal dimension to the whole of the space. It is shown that these conditions are satisfied for the n-point action of the pure Lorentz group and for a standard action of the Lorentz group of arbitrary signature.

On the algebraic structure of the unitary group.

Éric Ricard, Christian Rosendal (2007)

Collectanea Mathematica

We consider the unitary group U of complex, separable, infinite-dimensional Hilbert space as a discrete group. It is proved that, whenever U acts by isometries on a metric space, every orbit is bounded. Equivalently, U is not the union of a countable chain of proper subgroups, and whenever E ⊆ U generates U, it does so by words of a fixed finite length.

On the rigidity of webs

Michel Belliart (2007)

Bulletin de la Société Mathématique de France

Plane d -webs have been studied a lot since their appearance at the turn of the 20th century. A rather recent and striking result for them is the theorem of Dufour, stating that the measurable conjugacies between 3-webs have to be analytic. Here, we show that even the set-theoretic conjugacies between two d -webs, d ≥ 3 are analytic unless both webs are analytically parallelizable. Between two set-theoretically conjugate parallelizable d -webs, however, there always exists a nonmeasurable conjugacy; still,...

Reedy categories which encode the notion of category actions

Julia E. Bergner, Philip Hackney (2015)

Fundamenta Mathematicae

We study a certain type of action of categories on categories and on operads. Using the structure of the categories Δ and Ω governing category and operad structures, respectively, we define categories which instead encode the structure of a category acting on a category, or a category acting on an operad. We prove that the former has the structure of an elegant Reedy category, whereas the latter has the structure of a generalized Reedy category. In particular, this approach gives a new way to regard...

Semigroup actions on tori and stationary measures on projective spaces

Yves Guivarc'h, Roman Urban (2005)

Studia Mathematica

Let Γ be a subsemigroup of G = GL(d,ℝ), d > 1. We assume that the action of Γ on ℝ d is strongly irreducible and that Γ contains a proximal and quasi-expanding element. We describe contraction properties of the dynamics of Γ on ℝ d at infinity. This amounts to the consideration of the action of Γ on some compact homogeneous spaces of G, which are extensions of the projective space ℙ d - 1 . In the case where Γ is a subsemigroup of GL(d,ℝ) ∩ M(d,ℤ) and Γ has the above properties, we deduce that the Γ-orbits...

Shadowing in actions of some Abelian groups

Sergei Yu. Pilyugin, Sergei B. Tikhomirov (2003)

Fundamenta Mathematicae

We study shadowing properties of continuous actions of the groups ℤ p and ℤ p × ℝ p . Necessary and sufficient conditions are given under which a linear action of ℤ p on ℂ m has a Lipschitz shadowing property.

Théorème de Kurosh pour les relations d’équivalence boréliennes

Aurélien Alvarez (2010)

Annales de l’institut Fourier

En théorie des groupes, le théorème de Kurosh est un résultat de structure concernant les sous-groupes d’un produit libre de groupes. Le théorème principal de cet article est un résultat analogue dans le cadre des relations d’équivalence boréliennes à classes dénombrables, que nous démontrons en développant une théorie de Bass-Serre dans ce cadre particulier.

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